Group Theory

Binary operation on sets:

A binary operation on a set S is a mapping (function) from SxS to S.  As function, a binary operation should map an element in SXS to one and only one element in S.  Generally, these binary operations will be denoted by ‘*’.

* :SXS→S

Equivalently, for any (a, b)∈SXS there exist an element c∈S such that, a*b = c.

Example:

The usual multiplication ‘x’ on the set of natural numbers N= {1, 2, 3, …} is a binary operation.  Symbolically,

x :NXN→N

and for any pair of numbers we may find their multiplication which is a number in N.

Definition of Group:

A set G together with a binary operation ‘*’ defined on it is said to be a group if it satisfies the following axioms.
Closure property:

There must exist an element x*y ∈ G, for every x, y ∈ G

Associative property:

x*(y*z) = (x*y)*z, for every x, y, z ∈G

Identity property:
There exist an element e∈G such that,

e*x = x*e = x, for every x∈G

Inverse property:
There exist an element x-1∈G for every x∈G such that,

x*x-1 = x-1*x = e

Then the group G under * is denoted by the pair (G, *).
Abelian Group or Commutative Group:
A group (G, *) which in addition satisfies the commutative law,

x*y = y*x, for every x, y∈G

is said to be an abelian group or a commutative group.

More on Groups:

A set G together with a binary operation ‘*’  defined on it is said to be a semi-group if it satisfies only Closure and Associative properties of Groups.

A set G together with a binary operation ‘*’  defined on it is said to be a monoid if it satisfies Closure, Associative and Identity  properties of Groups.

The group generated by ‘a’ such that,

<a> = {a, a*a, a*a*a, …} = {an, for any integer n} (multiplicative group) <a> ={a, a+a, a+a+a, …}={na, for any integer n} (additive group)

is said to be the Cyclic group generated by ‘a’ and ‘a’ is said to be the generator of <a>.

Example:

Consider the set A = {i, -i, 1, -1}. Take a =i then i2 = -1, i3 = -i, i4 = 1. Then we may write, <i>={i, -i, 1, -1}. Easily we may prove that (A,*) is a group.  This group is said to be the cyclic group generated by i whose order is 4 since i4 = 1, the multiplicative identity and ‘i’ is said to be the generator of <i>.
A group (G,*) is said to be a cyclic group if for an element ‘a’ ∈G, G is equal to the group <a> , generated by ‘a’.

i.e., G = <a>, for some a∈G

n is said to be the order of G if an=e, the identity element.

A subset H of a group (G,*) is said to be a subgroup of G if (H, *) itself forms a group.  Equivalently, if H ⊂ G, and H satisfies Closure, Associative, Identity and Inverse properties under the same binary operation ‘*’, under which G is group then (H, *) is said to be a subgroup of (G,*).

Left Coset:
Let (H,*) be a subgroup of a group (G,*).  Then for some x∈G, the set given by,

xH={xh|h∈H}

is said to be the left coset of H in G.
Right Coset:
Let (H,*) be a subgroup of a group (G,*).  Then for some x∈G, the set given by,

Hx={hx|h∈H}

is said to be the right coset of H in G.

A subgroup (N,*) of the group (G,*) is said to be a normal subgroup of G iff

gng-1∈N, ∀g∈G and ∀n∈N

Homomorphisms of Groups:

Let (G,*) and (G1, ⊛)  be groups.  A mapping f between G and G1 is said to be a homomorphism if f preserves operations on groups. That is,

if f:G→G1 satisfies f(x*y) = f(x)⊛f(y), ∀x,y∈G

then f is said to be a homomorphism between G and G1.  Equivalently, a mapping which maps ‘multiplication of elements in G’ to ‘the multiplication of their images in G1‘ is said to be a homomorphism between G and G1.

Isomorphism of Groups:

Let (G,*) and (G1, ⊛)  be groups.  A mapping f between G and G1 is said to be an isomorphism if,

  1. f is 1-1
  2. f is onto and
  3. f is a homomorphism.

Two groups (G,*) and (G1, ⊛) are said to be isomorphic if there exist an isomorphism between G and G1 .

Automorphism of Groups:

An isomorphism of a group (G,*) into itself is said to be an automorphism.

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